Math 152
Sample Problems for material covered after exam 3

1.

Calculate the following integrals using the substitution method.

(a)

$\int_0^2{\frac{1}{(2x-5)^3}dx}$

(b)

$\int \frac{x^4}{1+x^5}^2dx$

(c)

$\int_0^{\sqrt{\pi}} x\sin{3x^2}dx$

(d)

$\int_3^7\frac{\ln x}{x}dx$

2.

Calculate the following integrals using partial fractions.

(a)

$\int \frac{1}{x(x-2)}dx$

(b)

$\int_1^2 \frac{x-2}{x^2+x}dx$

3.

Calculate the following integrals using integration by parts.

(a)

$\int x^{-1/2}\ln x dx$

(b)

$\int x e^{2x} dx$

(c)

$\int x \sin x dx$

4.

Solve the differential equation $y'(x)=\frac3 x -2\sqrt{x}+5$

5.

Solve the differential equation $y'=\frac{t^2}{y^2}$ with y(0) = 3

6.

At time $t=0\$ 300 miligrams of a drug is introduced into a patient's body. The drug is gradually eliminated and its amount A(t) decreases according to the equation

$$\frac{dA}{dt}=-0.017 A,$$

where time is measured in minutes.

(a)

Find the amount of the drug after 1 hour.

(b)

When should we administer the next shot if we do not want the amount of drug in the body to fall below 100 miligrams?

7.

Compute the amount of work done by a force $F(x) = \frac{1}{x^3}$ over the interval $a \leq x \leq b$. What happens to the amount of work done as $x \rightarrow \infty$?

8.

The density of sardines (number of fish per cubic meter) is given by a density function

$$d(x)=0.005x^2(82-x),\ 0\leq x\leq 82$$

where x is the depth of water.

(a)

Find the depth at which the density is maximal. What is the maximal density?

(b)

Find $d(0),\ d(82)$, inflection points (if any) and sketch the graph of d(x) for $0\leq x\leq 82$.

(c)

Find the number of sardines in the water column with cross section 1 m² in depth between 20 and 40 meters.